Lecture Notes in Logic, 47

ISBN-13: 9781107014527
554 pages. Hardcover.Buy now (when ordering, include the discount code ASL2016 to receive the 25% ASL member discount)

Descriptive complexity theory establishes a connection between the computational complexity of algorithmic problems (the computational resources required to solve the problems) and their descriptive complexity (the language resources required to describe the problems). This groundbreaking book approaches descriptive complexity from the angle of modern structural graph theory, specifically graph minor theory. It develops a ‘definable structure theory’ concerned with the logical definability of graph theoretic concepts such as tree decompositions and embeddings. The first part starts with an introduction to the background, from logic, complexity, and graph theory, and develops the theory up to first applications in descriptive complexity theory and graph isomorphism testing. It may serve as the basis for a graduate-level course. The second part is more advanced and mainly devoted to the proof of a single, previously unpublished theorem: properties of graphs with excluded minors are decidable in polynomial time if, and only if, they are definable in fixed-point logic with counting.

Table of Contents

Introduction

Part I The basic theory

Background from graph theory and logic

Descriptive complexity

Treelike decompositions

Definable decompositions

Graphs of bounded tree width

Ordered treelike decompositions

3-Connected components

Graphs embeddable in a surface

Part II Definable decompositions of graphs with excluded minors

Quasi-4-connected components

K5-minor free graphs

Completions of pre-decompositions

Almost planar graphs

Almost planar completions

Almost embeddable graphs

Decompositions of almost embeddable graphs

Graphs with excluded minors

Bits and pieces

Appendix Robertson and Seymour’s version of the local structure theorem